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Theorem ssint 4302
Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint
Distinct variable groups:   ,   ,

Proof of Theorem ssint
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfss3 3493 . 2
2 vex 3112 . . . 4
32elint2 4293 . . 3
43ralbii 2888 . 2
5 ralcom 3018 . . 3
6 dfss3 3493 . . . 4
76ralbii 2888 . . 3
85, 7bitr4i 252 . 2
91, 4, 83bitri 271 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  e.wcel 1818  A.wral 2807  C_wss 3475  |^|cint 4286
This theorem is referenced by:  ssintab  4303  ssintub  4304  iinpw  4419  trint  4560  oneqmini  4934  fint  5769  sorpssint  6590  iscard2  8378  coftr  8674  isf32lem2  8755  inttsk  9173  isacs1i  15054  mrelatglb  15814  fbfinnfr  20342  fclscmp  20531  dfrtrcl2  29071  fneint  30166  topmeet  30182  igenval2  30463  ismrcd1  30630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3482  df-ss 3489  df-int 4287
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